This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions
The hypocenter of a group is defined as the limit of the lower central series of the group, that is, the intersection of all members of the lower central series.
The hypocenter of any subgroup is contained in the hypocenter of the whole group.
This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once
The hypocenter of the hypocenter is the hypocenter. The image cum fixed-points of this are the hypocentral groups.
This subgroup-defining function is quotient-idempotent: taking the quotient of any group by the subgroup, gives a group where the subgroup-defining function yields the trivial subgroup
View a complete list of such subgroup-defining functions
The hypocenter of the quotient of a group by its hypocenter is trivial.
The hypocenter of any group is a fully characteristic subgroup.